<p>We consider the three-particle Schrödinger operator <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1737_Article_IEq1.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="56" /> </InlineMediaObject> <EquationSource Format="TEX">$H_{\mu }(K),\,$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>H</mi> <mi>μ</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>K</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> <mspace width="0.166667em" /> </mrow> </math></EquationSource> </InlineEquation> <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1737_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="127" /> </InlineMediaObject> <EquationSource Format="TEX">$ K\in \mathbb {T}=(-\pi ,\pi ],$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>K</mi> <mo>∈</mo> <mi mathvariant="double-struck">T</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mo>-</mo> <mi>π</mi> <mo>,</mo> <mi>π</mi> <mo stretchy="false">]</mo> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> associated with a system of three fermions with potential interacting neighboring sites on one dimensional lattice <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1737_Article_IEq3.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb {Z},$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="double-struck">Z</mi> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> where <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1737_Article_IEq4.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">$\mu $</EquationSource> <EquationSource Format="MATHML"><math> <mi>μ</mi> </math></EquationSource> </InlineEquation> is energy of interaction of two fermions. We prove that for sufficiently large <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1737_Article_IEq4.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">$\mu $</EquationSource> <EquationSource Format="MATHML"><math> <mi>μ</mi> </math></EquationSource> </InlineEquation> the operator <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1737_Article_IEq6.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="43" /> </InlineMediaObject> <EquationSource Format="TEX">$H_{\mu }(0)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>H</mi> <mi>μ</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> has a simple eigenvalue below the essential spectrum.</p>

Discrete Spectrum of the Schrödinger Operator of a System of Three Fermions on a One Dimensional Lattice

期刊论文
发表日期：

2025年7月21日

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J. I. Abdullaev,
Sh. H. Ergashova

摘要

We consider the three-particle Schrödinger operator $H_{\mu }(K),\,$ $H_{μ} (K),$ $ K\in \mathbb {T}=(-\pi ,\pi ],$ $K \in T = (- π, π],$ associated with a system of three fermions with potential interacting neighboring sites on one dimensional lattice $\mathbb {Z},$ $Z,$ where $\mu $ $μ$ is energy of interaction of two fermions. We prove that for sufficiently large $\mu $ $μ$ the operator $H_{\mu }(0)$ $H_{μ} (0)$ has a simple eigenvalue below the essential spectrum.